an optimal gear shifting strategy for heavy trucks with

An Optimal Gear Shifting Strategy for Heavy Trucks with Trade-off Study between Trip Time and Fuel Economy

X.X. Zhao1*, A. H. Ing2, N. L. Azad2 and J.McPhee2

1 University of Science and Technology Beijing School of

Mechanical Engineering, XueYuan Street 30, Beijing 100083,

China 2 University of Waterloo- Department of Systems Design

Engineering, 200 University Avenue West, Waterloo N2L3G1,

Canada

E-mail: [email protected]




*Corresponding author

Abstract: Advances in vehicle technology have improved

driving performance and fuel efficiency. Further increases in fuel

efficiency of a heavy mining truck can be achieved by optimizing

gear shifting strategy. Using characteristic tests of the diesel engine,

a high-fidelity model of a mining truck was built in MapleSim and

a consistent low-order model was developed in Matlab. The

dynamic programming approach has been applied to the low-order

model of the specialized off-road 30-tonne mining truck over a

fixed route in a mining area. There were two competing objectives:

fuel use and trip time. Weighting coefficients were used to

combine these objective functions to a single value, and a Pareto

curve was created to analyse the effect of the weights on the fuel

use and trip time. Using the control strategy obtained from

dynamic programming on the high-fidelity model, it is estimated

that forty-thousand litres of fuel can be saved annually based on a

mine which produces 110 kilotons of coal per day.

© Inderscience Enterprises Ltd. The original source of publication is available at InderScience. Zhao, X. x., Ing, A. h., Azad, N. l., & McPhee, J. (2015). An optimal gear-shifting strategy for heavy trucks with trade-off study between trip time and fuel economy. International Journal of Heavy Vehicle Systems, 22(4), 356–374. https://doi.org/10.1504/IJHVS.2015.073205

https://doi.org/10.1504/IJHVS.2015.073205

An Optimal Gear Shifting Strategy for Heavy Mining Trucks with Trade-off

Study between Trip Time and Fuel Economy

Keywords: dynamic programming; heavy mining truck; gear

shifting strategy; Pareto curve.

Biographical notes: Xinxin Zhao is a PhD student in the school of

mechanical engineering at the University of science and

technology Beijing, China. Her research interests are in the field of

motion dynamics and gear shifting control.

Adam Hu Ing has graduated from the department of systems

design engineering at University of Waterloo. His research

involves the parallel hybrid electric vehicle research project.

Nasser Lashgarian Azad is an Assistant Professor in Systems

Design Engineering at University of Waterloo. His primary

research interests lie in modelling, estimation and control of

complex dynamic mechanical and multi-domain physical systems,

with special emphasis on advanced modelling and model reduction

methods, sensitivity analysis techniques, nonlinear and optimal

control, with applications to advanced vehicle systems, such as

modern automotive powertrains and vehicle dynamics control

systems.

John McPhee is a Professor in Systems Design Engineering at the

University of Waterloo, Canada, and the Canada Research Chair in

System Dynamics. His main area of research is multibody system

dynamics, with principal application to the analysis and design of

vehicles, mechatronic devices, biomechanical systems, and sports

equipment. He was the Executive Director of the Waterloo Centre

for Automotive Research until 2009, after which he spent a

sabbatical year at the Toyota Technical Centre in Ann Arbor,

Michigan. He is a Fellow of the American Society of Mechanical

Engineers and the Canadian Academy of Engineering.

1 Introduction

Heavy trucks are used in mining operations to move dirt and ore.

These payloads often weigh up to 100 tons. The trucks operate for over 8

hours daily, and thus consume significant amounts of fuel during their

operating lifetime. Since oil is a non-renewable resource and becoming

more expensive, operational costs can be reduced by improving driving



efficiency. One method to reduce fuel consumption is to replace the diesel

powertrain with a hybrid electric powertrain; however, it is very expensive

to replace an entire fleet of mining trucks. A more cost-coefficient

approach to save fuel over a known route is to optimise the vehicle gear

shifting strategy.

The optimal gear shifting strategy will reduce the fuel consumption

by selecting the most efficient gear. It can be obtained using a

mathematical model of the powertrain and an optimization algorithm that

will find the optimal control law.

Two types of models are used in this work: high fidelity and low-order

models. High-fidelity models capture most of the dynamics of the system

but solve slowly. One popular powertrain model is the Mean Value

Powertrain model, which contains the Mean Value Engine Model (MVEM)

(Saerens et al, 2009). Low-order models solve quickly, but may lose many

details in the system. Often, low-order models are used to derive a control

strategy instead of a high-fidelity model because of the computational

efficiency (Fu and Bortolin, 2012).

Low-order models are often called control-oriented model because they

are built for control design approaches, such as Pontryagin’s minimum

principle (Chang and Morlok, 2005) and dynamic programming (Mensing

et al., 2013; Hellström et al, 2006; Liamas et al, 2013).

Dynamic programming requires a control-oriented model of the

vehicle, and all trip information. In the past, dynamic programming has

been formulated to minimize fuel consumption for passenger cars (F.

Xavier L.C., 2012), city buses (Nouveliere et al., 2008) and heavy trucks

(Fu and Bortolin, 2012, Fröberg et al., 2006; Hellström et al, 2009;

Hellström et al, 2010). Dynamic programming has also been formulated to

minimize both trip time and fuel consumption. It is utilized to carry out a

fuel-optima gear shift map and a throttle control problem over many

different acceleration profiles (Kim et al., 2007). The optimal gear shift

strategy for passenger vehicles equipped with an AMT has been designed

by using dynamic programming (Ngo et al., 2013). Simulation results over

the standard driving cycles represent a substantial improvement in the fuel

economy. Since there are not standard driving cycles for the heavy duty

truck, the optimal gear shift strategy is extracted based on the look-ahead

road conditions. Besides, the look-ahead information can be used for

improving fuel efficiency. There is a device exhibiting the gear shifting

strategy respect to driving condition data such as vehicle mass, driving

resistance and topography data(Sauter et al, 2009).

A control-oriented model and a high-fidelity model of the vehicle

powertrain, which are consistent with each other, are developed. The



high-fidelity model includes: a diesel engine, automatic transmission,

drive shaft, final drive and wheels. The fuel consumption model is based

on experimental data for the engine in the baseline vehicle, which is a

30-tonne mining truck. The road topography for a route in North-Western

China is collected beforehand, and the gear shifting and vehicle velocity

are optimized for this trip. By tuning the weighting coefficients associated

with fuel consumption and time in the objective function, driving

behaviour can be obtained which balances travel time and fuel

consumption. Using a high-fidelity model, it’s shown that the gear shifting

strategy improves both the fuel consumption and travel time.

The next section describes the low and high-fidelity models of vehicle

and the comparison between two models with given gear shifting strategy.

In the third section, the optimal control law for a known route is obtained

from the dynamic programming approach. The fourth section shows the

result of a trade-off study applied to the high-fidelity model, and the

model in the loop simulation is shown.

2 Modelling

Two models are presented in this section: a high-fidelity model and a

low-order model. First the similarities in both models are described then

the differences are highlighted.

A mining truck powertrain usually consists of a diesel engine,

automatic transmission, and final (output) drive. The final drive is

assumed to be a fixed gear ratio with no power losses. The vehicle wheels

are assumed to have no slip. Figure 1 depicts the inputs and outputs of

each subsystem of the driveline in a mining truck, where Mx is the torque

and Φx represents the angle.

Figure 1 Schematic of a mining truck powertrain



2.1 Vehicle longitudinal dynamics

Both the high-fidelity and low-order model assume only the

longitudinal dynamics of the truck is important. The equations for

longitudinal dynamics can be found in the following equations (Guzzella

and Sciarretta, 2007, Ivarsson et al., 2009). The frictional forceLF is the

sum of air drag, rolling resistance, and gravitational force down the sloped

road. The torque due to friction force and torque balance formula of wheel

is shown in Eq. 1 and 2 and the slope of the road is represented by . The

other parameters can be found in Table 1.

sincos2

11

23 rmgrmgcrACrF rwalairL (1)

LLwww MrFMJ (2)

Table 1 Model parameters

Parameter Symbol Value Unit Parameter Symbol Value Unit

Vehicle mass m 24500 [kg] Final drive if 12.5

Wheel radius r 0.734 [m] Air density ρ 1.21 [kg/m3]

Frontal area Af 7 [m2] Gear 1 ratio i1 5.182

Air drag

coefficient

Cair 0.7 Gear 2 ratio i2 3.188

Rolling

resistance

coefficient

cr1 0.03 Gear 3 ratio i3 2.021

Wheel inertia Jw 5 [kgm2] Gear 4 ratio i4 1.383

Engine

inertia

Je 1.54 [kgm2] Gear5 ratio i5 1

Cylinders ncyl 4

2.2 High Fidelity Powertrain Model

This is based on the equations given in (Chan et al., 2007), and the

high-fidelity model is a forward quasi-static model regarding the throttle

angle as input and mass of fuel as output. MapleSim with the Driveline

library is used to model the mining truck powertrain. Parameters used in

the mean value engine model with turbocharger are shown in Table 1 and

the Nomenclature. Mining trucks have 4 planetary gears which can

produce 5 discrete ratios by (dis)engaging clutches. The MapleSim model

of an automatic transmission is introduced here.



2.2.1 Mean value engine model

The engine is modeled using the Mean Value Engine Model (MVEM),

which is a complex model that captures physical dynamics such as fuel

mass flow rate, pressure, and temperature. Diesel engines often use

turbochargers for extra power. Shown in Figure 2, The MVEM breaks

down the engine model into sub-models: compressor, turbine, intake

manifold, exhaust manifold, and combustion and torque production.

Figure 2 A schematic diagram of a turbocharged diesel engine

The turbine and compressor are mechanically connected; when air

from the exhaust moves past the turbine, it causes the turbine to spin,

subsequently spinning the compressor. The compressor compresses the

entering air which flows into the intake manifold. As shown in equation 3,

the first law of thermodynamics is used to relate the compressor torque

cM to the mass flow through the compressor. The reader is referred to the

Nomenclature section at the end of the paper for definitions of the other

variables and parameters.

1

1

amb

cx

tcc

ambpac

cp

pTCmM

(3)

The intake manifold ensures the engine cylinders get sufficient air for

combustion. The mass flow rate for the intake manifold is described by

the ideal gas law Eq. 4.

dt

dp

RT

Vmmm im

im

imiecim (4)

A turbocharged diesel engine operates differently than a naturally

aspirated gasoline engine. At the end of the compression stroke, the

diesel engine injects fuel into the engine cylinder just before the desired



start of combustion. The expansion of gases (hot air and emissions) from

the combustion process pushes against the cylinder to create mechanical

power. Seen in Eq. 5, the mass flow rate is dependent on the mass of

fuel per cylinder and per stroke (uf) and number of engine strokes.

r

cylef

fn

num

(5)

Derived from the definition of indicated thermal efficiency, the

indicated power indP

is :

indlhvfind QmP

(6)

The engine rotational dynamics model is a simple torque balance:

eeloadfind JMMM (7)

The difference between the turbine exhaust gas pressure and the

atmosphere is not ignored. The drop in gas pressure from crossing the

exhaust manifold is shown in Eq. 8 (Yanakiev and Kanellakopoulos, 1995).

)(42

10

22

eximeximimex TTkmppp (8)

The rotational dynamics of the turbine-compressor shaft are described

by Euler’s equation, where the moment of inertia of turbocharger rotor is

denoted by tcJ :

tctcct JMM (9)

These equations for the MVEM with turbocharger are used in the

high-fidelity model and implemented in MapleSim.

2.2.2 Gearbox

The automatic gearbox in a mining truck consists of multiple planetary

gears connected together. This has the advantage of a large gear ratio,

reduced weight, and high torque output. In total, there are 4 planetary

gearsets and 6 clutches. The structure of the gearbox can be found in

Figure 3, where P, B and C represent the planetary set, brake and clutch

respectively.

Figure 3 Gearbox model in MapleSim



In the high-fidelity model, the gearbox considers the inertia of each

component (represented by the rectangular block in Figure 3). As the

number of teeth and inertia for each gear is known, the dynamic model of

planetary gear train is easily obtained from the equations for a planetary

gearset, shown in Eq. 10-13, where α, β, γ and θ is the angular

displacement of sun gear, carrier, pinion and ring gear. RSn, RCn, RPn and

RRn represent the radius of sun gear, carrier, pinion and ring gear

respectively, while n means the number of planetary.

1111111

111111

)2( RPPS

SPS

RRRR

RRR

(10)

2222221

212221

)2( RPPS

SPS

RRRR

RRR

(11)

3333331

323331

)2( RPPS

SPS

RRRR

RRR

(12)

4444441

424441

)2( RPPS

SPS

RRRR

RRR

(13)

To build the powertain system dynamic model with Lagrange method,

the kinetic energy of gear box is presented by Eq. 14. The box consists

with 10 components since some parts connect each other. Based on the

mesh equations of gear, there are 10 variables and 8 constraint equations.

The other 6 variables could be presented regard to input angular velocity

2 and output one 1

. With visual work principle, the system visual work

is shown by Eq. 15, where An is the radius ration and explained in

Nomenclature.

4433234223411112 RPRPRSPCRPCS

(14)



2443323421112

14433234121112

)(

)]1()1()1()1([

ATATTATAT

ATATTATATW

RRRSCS

RRCRCS

(15)

The Lagrange equations of system are shown in Eq.16-17,

1

1

1 :

Q

dt

dequation

(16)

2

2

2 :

Q

dt

dequation

(17)

Taking the partial derivatives of the kinetic energy in Eq. 14 with respect

to 2 and 1

and substituting them into Eq. 16 and 17 along with the

generalized moments from Eq. (15) yield 2 and 1

equations of

dynamics for the planetary gear set (Eq.18 and 19). The parameters of

planetary gear set are shown in Table2.

)1()1()1()1( 4433234121112212111 ATATTATATDD RRCRCS

(18)

443323421112222121 ATATTATATDD RRRSCS

(19)

Table 2 Planetary gear set parameters

Planetary No.

No. Parameter

Radius

(mm)

Inertia

(kg/m2)

Gear teeth Planet pinion No.

P1

Sun gear 55.4 1.25*10-2 48

6 Ring gear 92.35 2.41*10-1 80

Pinion 18.45 4.81*10-5 16

P2

Sun gear 35.8 - 31

4 Ring gear 96.7 1.7*10-1 81

Pinion 27.7 2.99*10-4 24

P3

Sun gear 53.3 - 42

4 Ring gear 116.85 1.73*10-1 95

Pinion 31.75 4.21*10-4 25

P4

Sun gear 27.9 - 22

4 Ring gear 116.85 1.73*10-1 92

Pinion 44.85 2.82*10-3 35



2.3 Low-order Powertrain Model

The low-order model is used to determine the optimal control law,

because it solves quickly. The low-order model is a steady-state

backwards model that the only input is the throttle angle (Chan et al.,

2007), and thus, inertial effects are ignored and the system is assumed to

be able to reach any desired behaviour instantly.

2.3.1 Engine

The engine fuel consumption model is based on empirically collected

data. The data is stored in a map, which uses torque and speed to

interpolate the fuel consumption rate. As shown in Eq. 20, the engine

speed is a function of the vehicle speed W and gearbox ratio. The mass

of fuel is calculated by using the following interpolation function ef (Eq.

21), which is related with torque of engine ICE and angular speed of

engine ICE . Combining the measurement data of engine, the rate of fuel

consumption fICEm ,

can be plotted in Figure 4.

RWICE ii 0 (20)

ICEICEefICE fm ,, (21)

Figure 4 Fuel mass flow rate of a diesel engine based on experimental

data

2.3.2 Gearbox

The gearbox is modelled as a simple gear ratio with no losses. The

equations for the gearbox are given by:



RAB i (22)

RBA i (23)

where, A is the input and B is the output of the gearbox, represents the

torque, is angular velocity and Ri (R=1,2,…,5) is the gear ratio as

shown in Table 1.

3 Optimal Gear Shifting Strategy

3.1 Trip Information

Optimization was used to determine the optimal gear shifting control

law for the mining truck. Dynamic programming is an effective method

for finding the optimal gear shifting strategies for a trip will that is fully

defined beforehand. The control inputs of the mining truck can be

formulated as an optimization problem, which can be solved by dynamic

programming.

The vehicle trip data was gathered using Google Earth (Google Earth,

2014). It is approximately 3.4 km on a colliery in North-western China. As

shown in Figure 5, the route starts at the star at the top, and ends at star at

the bottom. The altitude and slope of the route was extracted at discrete

intervals, then smoothed using high-order interpolation, as shown in Fig

5(b).

Figure 5 Trail road topography and estimated road slope

(a)



(b)

3.2 Objective Function

The problem is formulated in a way that minimizing fuel consumption

and/or trip time can be achieved by changing two weighting parameters:

1 for fuel use and 2 for trip time. These parameters weigh the objective

score to minimize trip time, fuel consumption, or both. The cost function

at each instant k is:

1,...,2,1,02,1 Nktm kkfk (24)

where the N is the number of steps of this optimal problem.

3.3 Control Inputs and Constraints

As the engine is mechanically connected to the wheels in the

low-fidelity model (Eq. 20), its speed is a function of the vehicle speed

and the gear ratio. The engine has a minimum idle speed of 750(rpm) and

maximum speed of 2350(rpm); therefore each gear ratio in the

transmission has a corresponding minimum and maximum speed, which

are shown in Table 3.

The gear change input is a function of the current gear. Gear changes

are only done in increments of +1, 0, or -1. If the current gear is the

highest gear there are two admissible signals: 0 or -1. Conversely if the

current gear is the lowest gear there are two admissible signals: 0 or +1.



Table3 The vehicle speed at different gear ratios

Gear Minimum speed

[Km/h]

Maximum speed

[Km/h]

1 3.2 10.03

2 5.2 16.31

3 8.21 25.72

4 11.99 37.6

5 16.59 51.9

An optimal control problem consists of an objective function, discrete

state expressions with constraints, and control signals. The low-order

powertrain optimal control problem is implemented in Matlab and can be

found in Eq. 25 and 26. It represents the relationship between the current

states at instant k and the states in last instant k-1.

Min

1

0

N

kkNJ (25)

Subject to constraints :

11,1

111,

1

1

1,,

111,

1

1

1

,

,

kkgkk

kkkf

k

k

kfkf

kkkw

k

k

kk

gugg

slopegmv

hmm

slopegv

hvv

6)

3.4 Dynamic Programming Algorithm

Dynamic programming tests all feasible control actions for each state in

stage k and chooses the control input which minimizes the cost function.

For this type of optimal control problem, dynamic programming often

starts at the end of the trip and proceeds backwards (Kirk, 2004). The

specific dynamic programming for this deterministic multi-stage decision

process will be expressed as the following steps and dynamic

programming parameters are listed in Table 4.



Table4 Dynamic programming parameters

Function Parameter Value Unit

Number of steps N 680

Step length h 5 m

Horizon hN 3400 m

Velocity

discretization v 5 hkm /

Lower engine bound idle, 750 rpm

Upper engine bound max,e 2350 rpm

Initial velocity 0v 15 hkm /

Initial gear 0g 2

The algorithm is briefly described below:

1. Set terminal cost to zero, 0NN x

2. Let 1 Nk

3. Solve the objective function to minimize cost, kgkkkkgkk

ukk uxfJuxxJ

kg

,1, ,,min,

where linear

interpolation is used to calculate objective scores that fall between

the discredited grids of states.

4. N = N-1. Go to step 2.

5. When N = 0, the optimal control law can be found starting at *

0x

and solving optimal control problem forwards.

3.5 Comparison of Models with Dynamic Programming

To verifying the dynamic programming algorithm used in low-order

model, the optimized shifting result was used in high-fidelity model. The

models are tested over a 3400-meter route which mentioned in 3.1. It starts

from second gear and ends at fourth gear which is shown in Figure 6(a).



Figure 6 Comparison of results between the low-order model and

high-fidelity model

As shown in Figure 6(b)-(d), response of the high-fidelity and low-order

model is compared, with both models having the same inputs. As can be

seen from Figure 6(b) and (c), the results obtained from high-fidelity

model, trip time and fuel consumption, are consistent with the solution of

the low-order model. The control oriented model has a longer trip time

around 294s than the high-fidelity model. The total fuel consumed over

the trip is 3.21L for the high-fidelity model and 3.27L for the low-fidelity

model, which is an error of 1.8%.

In figure 6(d), the vehicle speed of the high-fidelity model is slightly

faster during 105m to 200m, thus a shorter trip time and less fuel

consumed is expected. In summary, the agreement between models is

acceptable for the low-fidelity model to be used to find the optimal control

law for the high-fidelity model.

4 Results and Discussion

4.1 Tradeoff study and explanation

The aim of optimizing gear shifting strategy is to minimize fuel

consumption without significantly increasing trip time. The trade-off

between trip time and fuel consumption was examined by using different

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

(a) Distance [m]

Gear

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300

(b) Distance [m]

Trip tim

e [s]

High-fidelity model

low-order model

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3

3.5

(c) Distance [m]

Fuel [L

]

High-fidelity model

low-order model

0 500 1000 1500 2000 2500 30000

10

20

30

40

50

(d) Distance [m]

Velo

city [km

/h]

High-fidelity model

low-order model



weights of 1 and

2 while solving the optimal control problem. The

results can be found in Figure 7.

Figure 7 demonstrates the trade-off between trip time and fuel use with

varying weight coefficients. If only the trip time is weighted, the trip is

the shortest and uses the most fuel. When only the fuel consumption is

weighted, the trip is the longest, but does not have the lowest fuel

consumption. When only fuel consumption is weighted (i.e. weights of

[1,0]), one would expect the lowest fuel consumption. However, the

lowest fuel consumption can be found with mixed weights of [0.4, 0.6].

The minimum fuel consumption can be found when 1 and

2 are 0.4 and

0.6 respectively. For fuel consumption is related to trip time, the case with

two weight include is a trade-off between trip time and fuel usage.

Figure 7 Pareto curve of weight coefficients

The shape of the curve in Figure 7 is unintuitive. Figure 8 is the ratio of

fuel consumption to trip time versus1 . Although the minimum fuel

consumption is obtained with a mixed weight, the minimum ratio appears

in the case where only fuel consumption is considered. (1 =1,

2 =0)

Figure 8 The ratio of fuel consumption and trip time



Setting weight parameters of objective function, three kind of gear

shifting strategies are shown in Figure 9. The optimization of the gear

shifting is presented for certain trip exhibited in Figure 6. By examining

the engine operating points in Figure 10, it can be seen that a slower

vehicle speed corresponding to longer trip time will cause the engine to

operate outside of its most efficient region. As the engine speed is a

function of vehicle speed, a mixed weighting will allow the engine to be

controlled in its most efficient region.

Figure 9 The different shifting strategies for three cases

Figure 10 Engine operation points. a) weighted for speed, b) mixed

weighting, c) weighted for fuel saving

0

1

2

3

4

5

6

0 1000 2000 3000

gea

r sh

ifti

ng s

tra

tegy

distance s:m

[λ1,λ2]=[1,0]

[λ1,λ2]=[0.4,0.6]

[λ1,λ2]=[0,1]



1000 1200 1400 1600 1800 2000 2200

200

400

600

800

1000

1200

1400

1600

Engine speed [rpm]

Engin

e t

orq

ue [

Nm

]

(a)[λ1,λ2]=[0,1]

191

195

203

215

231251

275 303335

Max torque [Nm]Fuel consumption [g/KWh]

* Dynamic programming

1000 1200 1400 1600 1800 2000 2200

200

400

600

800

1000

1200

1400

1600

Engine speed [rpm]

En

gin

e to

rqu

e [N

m]

(b)[λ1,λ2]=[0.4,0.6]

191

195

203

215

231251

275 303335



1000 1200 1400 1600 1800 2000 2200

200

400

600

800

1000

1200

1400

1600

Engine speed [rpm]

En

gin

e to

rqu

e [N

m]

(c)[λ1,λ2]=[1,0]

191

195

203

215

231251

275 303335





4.2 High-fidelity model simulation using optimal gear shifting strategy

The optimal gear shifting strategy obtained by dynamic programming

was used on the high-fidelity model of the mining truck, with fuel

consumption and trip time weights of 1 = 0.2, and

2 = 0.8 respectively.

The results from the high-fidelity model can be found below.

In this simulation, the results for trip time and fuel consumption with

the current strategy are 291.3s and 3.22L respectively. The current

strategy is rule-based and given from a transmission manufacturer. (Yu et

al., 2006) When compared to the current control strategy, the optimized

gear shifting strategy reduces trip time by more than 1.7% and fuel

consumption around 1% simultaneously.

As shown in Figure 11, the main differences between the two strategies

are at the beginning of the route. At 460 meters, the optimized strategy

downshifts earlier than the current strategy, to produce more torque for

uphill driving (see Fig.11). The optimal gear shifting strategy keeps it in

third instead of fourth gear during 200 meter to 265 meters. In Figure

10(c), the engine can be seen to be maintained in a high speed range where

it operates more efficiently.

Figure 11 Simulation results by different gear shifting strategy

0 500 1000 1500 2000 2500 30000

10

20

30

40

50

(a) Distance s: m

Velo

city k

m/h

Current strategy

Optimal control strategy

0 500 1000 1500 2000 2500 30001

2

3

4

5

6

(b) Distance [m]

Gear



According to the reported production of the Hei Daigou Mine, 110

kilotons per day of raw coal was produced (Zhun neng News, 2011).

Using a 30 ton truck, this would result in 3667 trips. Applying the

optimized gear shifting strategy to the road topology in this mine,

potentially forty-thousand litres of diesel can be saved per year.

5 Conclusion

This study shows the potential of an offline optimal controller on a

heavy truck to save both fuel and time by using prior knowledge of the trip.

A low-fidelity steady-state model and a high-fidelity quasi-static model of

a turbocharged heavy mining truck was developed. A custom gear shifting

strategy was applied to the two models and the results from the low order

model were compared with the high-fidelity model, which were very

consistent with each other. An objective function with weightings on the

fuel consumption and trip time was developed, and dynamic programming

was used to find the optimal control law which minimizes both. A Pareto

curve was used to determine the optimal weightings for the controller,

which reduced the fuel consumption and trip time by more than 1.7% and

1% respectively. Future work for the gear shifting controller is to validate

the results from the high-fidelity model against experimental data from a

real vehicle.

Nomenclature

2

21

S

R

R

RA

)( 112

212

RSS

RS

RRR

RRA

0 500 1000 1500 2000 2500 3000

1000

1500

2000

(c) Distance [m]

Engin

e s

peed [

rpm

]



3

3

3

R

S

R

RA

4

44

R

S

R

RA

2

1

15 A

R

RA

P

R

2

26

P

R

R

RA

3

37

P

S

R

RA

4

48

P

S

R

RA

paC Specific heat capacity of air at constant pressure, paC = 1011(J/kgK)

0k Coefficient for calculating the pressure

cm Mass flow rate of compressor (kg/s)

imm Mass flow rate of air at the intake manifold (kg/s)

iem

Mass flow rate of air from inlet manifold to engine (kg/s)

fm Mean mass flow rate fuel (kg/s)

exm Mass of fuel in exhaust manifold

cM Torque produced by the compressor (Nm)

fM Friction torque (Nm)

indM Engine indicated torque (Nm)

loadM Load torque (Nm)

tM Torque produced by the turbine (Nm)

rn

Number of revolutions per power cycle of an engine, rn = 2

ambp Ambient pressure, ambp = 101.35(kPa)

cxp Pressure at the compressor outlet (kPa)

exp Pressure in the exhaust manifold

imp Pressure at intake manifold inlet (kPa)

lhvQ Heating value of diesel, lhvQ = 6109.42 (J/kg)

R Gas constant for air, R = 287(J/kgK)

ambT Ambient temperature, ambT = 298.15(K)

exT Temperature of the exhaust manifold

imT Temperature at intake manifold (K)

fu Mass of fuel each cylinder and stroke (g/ (strokecylinder))



imV

Volume of the inlet manifold (m3)

tc Turbine angular velocity (rad/s), eeN

2

60(rpm)

c Efficiency of compressor

ind Thermal efficiency

References and Notes

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